© 1995 American Heart Association, Inc.
Articles |
Biphasic Restitution of Action Potential Duration and Complex Dynamics in Ventricular Myocardium
From the Departments of Physiology (M.W., R.F.G.) and Electrical Engineering (N.F.O.), Cornell University, Ithaca, NY.
Correspondence to Robert F. Gilmour, Jr, Department of Physiology, T8 023B VRT, Cornell University, Ithaca, NY 14853-6401. E-mail rfg2@cornell.edu.
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Abstract The purpose of this study was to determine whether biphasic restitution of action potential duration (APD) in ventricular muscle permits the development of complex dynamic behavior. Such behavior is expected because of the steep ascending slope of restitution and the presence of a maximum. Action potentials recorded from strips of epicardial muscle in which biphasic APD restitution occurred demonstrated a characteristic pattern of phase locking during progressive shortening of the pacing cycle length. 1:1 locking was replaced by irregular dynamics, which in turn was replaced by higher order periodic behavior (eg, 8:8 locking), then by 2:2 locking, and finally by 2:1 locking. Similar patterns of dynamic behavior were produced in a computer model by using a piecewise linear approximation of biphasic APD restitution. Features of APD restitution that were critical determinants of irregular dynamics included the slopes of the ascending and the nonmonotonic regions. These results suggest that rate-related alterations of APD and refractoriness may be affected significantly by small nonmonotonicities in APD restitution.
Key Words: ventricular arrhythmias • nonlinear dynamics • chaos
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Introduction |
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Rate-dependent alterations of action potential duration (APD) and refractoriness are believed to contribute importantly to the development of reentrant ventricular tachyarrhythmias.1 2 3 4 5 Consequently, the effects of constant pacing and sudden changes in pacing frequency on APD have been studied extensively (reviewed in Reference 66 ). Most often, APD has been described as a monotonic function of the stimulus coupling interval, both under steady state conditions and during restitution of APD (eg, see References 7 through 107 8 9 10 ). In some studies, however, restitution of APD was found to be biphasic; ie, as the stimulus coupling interval was prolonged, APD first increased to a local maximum, then decreased to a local minimum, and then increased to asymptotically approach a constant APD.7 11 12 13 14 15 16 Biphasic restitution has been found to be particularly prominent in human myocardium, where the difference in APD between the local maximum and minimum can be 25 to 30 ms.13 16 No consistent relation has yet been established between the presence of biphasic restitution and the region or species of heart examined or the presence or absence of myocardial disease.7 11 12 13 14 15 16 Similarly, the functional consequences of biphasic restitution with respect to rate-related changes in ventricular refractoriness are unknown.
Previous studies in Purkinje fibers have shown that under experimental circumstances in which the relation between total APD and the preceding diastolic interval was nonmonotonic, period-doubling bifurcations culminating in chaotic dynamics occurred.17 These dynamics are expected for any nonlinear system whose behavior can be described by a one-dimensional map that contains a steeply sloped region and a critical point (ie, a maximum or a minimum).18 Accordingly, it is expected that biphasic APD restitution, by virtue of its steeply ascending phase and presence of a maximum, would permit the development of complex dynamic behavior. The purpose of the present study was to determine whether, in fact, such behavior occurs. Because the characteristics of biphasic APD restitution could not be altered systematically or at will, an analytical model was developed to determine which features of biphasic restitution were most conducive to the generation of complex behavior.
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Adult dogs of either sex were anesthetized with Fatal-Plus (390 mg/mL pentobarbital sodium, 86 mg/kg IV, Vortex Pharmaceuticals), and their hearts were excised rapidly and placed in cool Tyrode's solution. Strips of epicardial muscle (2 to 5 mm wide, 5 to 10 mm long, and 1 mm deep) (n=12) were obtained from either ventricle. Strips of epicardial muscle were also obtained from 42- to 45-day-old canine fetuses (n=5) or from 80- to 103-day-old sheep fetuses (n=6). Before removal of the fetuses, maternal dogs were preanesthetized with atropine (0.15 mg SC) and acepromazine (3 mg SC) and subsequently were anesthetized with a combination of thiamylal (5 mg/lb of a 3% solution) and halothane (maintained at 1.5%). Maternal sheep were anesthetized with ketamine (1000 mg IV) and 4% halothane. All experimental procedures were approved by the Institutional Animal Care and Use Committee at Cornell University.
The preparations were superfused with normal Tyrode's solution at a rate of 15 mL/min. The composition of the Tyrode's solution (mmol/L) was MgCl2 0.5, NaH2PO4 0.9, CaCl2 2.0, NaCl 137.0, NaHCO3 24.0, KCl 4.0, and glucose 5.5. The Tyrode's solution was gassed with 95% O2 and 5% CO2 to produce a PO2 of 400 to 600 mm Hg, and the temperature was maintained at 37.0±0.5°C. The preparations were stimulated initially at a basic cycle length of 500 ms by using rectangular pulses of 2-ms duration and an intensity of 2.0 times the late diastolic threshold delivered via bipolar platinum wire electrodes insulated with polytetrafluoroethylene except at their tips (interelectrode distance, 1 mm). Transmembrane recordings were obtained by using standard techniques.19
The steady state relations between pacing cycle length and action potential amplitude (APA) and APD were determined by shortening the pacing cycle length from 500 ms to the cycle length at which 2:1 responses occurred. The behavior at each cycle length was considered to be stable if it persisted for >10 seconds. Because stimulation at short cycle lengths elicited action potentials during phase three of the preceding action potential, measurements of total APD required extrapolation of the slope of phase 3 at the time of activation. To avoid potential errors in the estimation of APD by use of such a method, the takeoff potential, defined as the membrane voltage at the moment of the delivery of the stimulus, was used as an index of the APD; ie, the more negative the takeoff potential, the shorter the preceding APD. The correspondence between takeoff potential and APD was confirmed by using the following protocol: Initially, the preparation was paced at a constant cycle length (S1S1, 150 to 300 ms), and the relation between the diastolic interval and the duration of the action potential elicited by a premature stimulus (S2) was determined for a range of S1S2 intervals. APD was determined by linear extrapolation of the terminal portion of phase 3 to the resting membrane potential. The protocol was then repeated, but with the addition of a second premature stimulus (S3) delivered at a fixed S2S3 interval. The relation between the diastolic interval and the takeoff potential of the response elicited by S3 was determined at each of the S1S2 intervals. The relations between diastolic interval and APD or takeoff potential were not significantly different for diastolic intervals less than zero (data not shown).
The time course of restitution of APA, APD, and latency (the time interval between delivery of the stimulus and the upstroke of the action potential) was determined as described previously.17 20 APD was measured at 90% of repolarization. APA, APD, and latency were then expressed as functions of the preceding diastolic interval, where the diastolic interval equaled the stimulus coupling interval minus the duration of the preceding action potential. Latency was measured because it has been shown to play an important role in determining complex dynamics in Purkinje fibers and in ventricular muscle exposed to heptanol.17 However, under the experimental conditions used for the present study, latency contributed minimally to the results.
Modeling of the Restitution Function
The relation between APD and the preceding diastolic interval
was modeled by using a piecewise linear approximation (12 linear
segments) of the experimental data. The dynamics of the restitution
function were analyzed by iterating the function at the desired pacing
cycle lengths and displaying the 950th to 1000th iteration for each
cycle length.
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Complex Dynamics in Ventricular Muscle
Shortening the pacing cycle length most often induced a transition from 1:1 to 2:2 to 2:1 stimulus-response locking. However, in 8 (4 of 12 adult canine, 2 of 5 fetal canine, and 2 of 6 fetal sheep) preparations, higher order periodicities and irregular dynamics were observed. As described below, the complex dynamics were associated with the presence of biphasic restitution of APD. To test whether inactivation of the transient outward current (Ito) at short coupling intervals contributed to the development of biphasic restitution, fetal tissue, which lacks Ito,21 was studied. The results obtained in the fetal and adult tissues were similar.
Examples of action potentials recorded during periodic and aperiodic
behavior and a segment of the bifurcation diagram are shown in Fig 1
. As the pacing cycle length was decreased from 200 to
135 ms, APD decreased, and 1:1 locking between the stimulus and APD was
maintained. Further shortening of the cycle length to 134 ms induced
irregular dynamics, which persisted as the cycle length was reduced to
130 ms. At a pacing cycle length of 129 ms, both 6:6 and 8:8 locking
occurred, which evolved into 4:4 locking as the pacing cycle length was
shortened from 128 to 123 ms. Pacing at cycle lengths of 122 and 121 ms
once again induced irregular dynamics. A 2:2 locking with increasing
alternans of APD occurred as the cycle length was shortened from 120 to
110 ms, culminating in 2:1 locking (ie, 2:1 block) at a cycle length of
109 ms.
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134 ms. The most negative
takeoff potentials correspond to the longest action potential
durations. See text for discussion.
To test whether the periodic and aperiodic behavior of the type
described in Fig 1
was derived from a one-dimensional deterministic
mechanism, the relations between successive APD and APA (ie, return
maps) were determined. Return maps for APA and for the equivalent of
APD (takeoff potential) during 4:4 locking and during aperiodic
behavior are shown in Fig 2. The return maps for APA
describe a steeply sloped nonmonotonic function that contains a
minimum, whereas the return maps for takeoff potential describe a
steeply sloped function that contains a maximum. The data points for
the aperiodic behavior are largely confined to a one-dimensional map,
as opposed to being distributed randomly, suggesting that they result
from the same mechanism as the periodic behavior.
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The underlying mechanism for the periodic and aperiodic behavior
was a biphasic relation between APD and the preceding diastolic
interval, as shown for higher order periodic behavior during constant
pacing (Fig 3A) and for restitution of APD using the
standard restitution protocol (Fig 3B). The "overshoot" of APD
(ie, the difference between the local maximum and the local minimum)
during restitution was 7 ms in this preparation and ranged from 2 to 22
ms in the other preparations.
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The relation between APA and the preceding diastolic interval was
monotonic during periodic behavior (eg, 8:8 locking; Fig 3A), as well
as during aperiodic behavior (not shown) and during restitution (Fig 3B). Accordingly, the nonmonotonic APA return map obtained during
aperiodic behavior (Fig 2C) reflected the nonmonotonic relation between
APD and diastolic interval rather than a nonmonotonic relation between
APA and diastolic interval. It is also worth noting that a wide range
of APA was associated with relatively small changes in APD during
periodic behavior (Fig 3A, where APD varied by 
20 ms, whereas APA
varied by 27 mV).
Analysis of Restitution Function Dynamics
A piecewise linear approximation of the APD restitution function
was derived from the experimental results shown in Fig 3B. Fig 4A shows the bifurcation diagram, minus transients,
obtained by iterating the function over a range of pacing cycle lengths
of 120 to 150 ms. As pacing cycle length was shortened from 150 to 137
ms, 1:1 locking was replaced by 2:2 locking and then by a region of
irregular dynamics between cycle lengths of 137 and 130 ms, followed by
stable 4:4 locking at a cycle length of 129 ms. The dynamics became
irregular again between pacing cycle lengths of 128 and 127 ms. Further
shortening of the cycle length induced 2:2 locking, which was replaced
by 2:1 locking at a cycle length of 122 ms.
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The results of the computer iterations mimicked the behavior seen
in the experimental preparation, in that they replicated the sequence:
1:1 locking followed by irregular dynamics, 4:4 locking, irregular
dynamics, 2:2 locking, and then 2:1 locking (Fig 1B). With respect to
differences between the results of the model and the corresponding
experiment (Fig 1B), the initial transition from 1:1 to 2:2 locking
observed in the model was not seen in the experiment. However, this
bifurcation was seen in other preparations, although it was difficult
to resolve, given the small changes in APD. In addition, the range of
cycle lengths over which the second region of 2:2 locking occurred (127
to 122 ms) was smaller in the model than in the experiment. This
discrepancy resulted from the fact that the model was based on the
experimentally determined restitution function, which did not include
electrotonic responses (Fig 3B), whereas the 2:2 locking observed
during constant pacing did include such responses.
The role of the overshoot of APD in the production of complex dynamics
was investigated further by (1) eliminating the overshoot, so that the
restitution function was monotonic (Fig 4B), and (2) increasing the
difference between the local maximum and local minimum from 7 ms (Fig 4A) to 11 ms (Fig 4C). In the absence of an overshoot, the dynamics
included only 1:1, 2:2, and 2:1 locking (Fig 4B). Increasing the
magnitude of the overshoot altered the transitions between periodic and
aperiodic regions and produced higher order periodic behavior (eg, 6:6
and 8:8 locking) (Fig 4C). In addition, both the range of cycle lengths
over which irregular dynamics occurred and the range of APD during
irregular dynamics increased.
Given the results shown in Fig 4, an analysis of the restitution
function was performed to identify the determinants of (1) the range of
APD that occurred during irregular dynamics and (2) the range of pacing
cycle lengths over which irregular dynamics occurred. With respect to
the first objective, the range of possible APD values for the aperiodic
behavior could be predicted from the restitution function by using the
following four equations: y=141, y=134,
y=f(basic cycle length-134), and y=f(basic cycle
length-141) (Fig 4A inset). The values 141 and 134 were the local
maximum ymax and local minimum
ymin of the restitution curve, respectively
(indicated by the horizontal lines in the figure). The equation
y=f(basic cycle length-134) defined the restitution
function f shifted to the left by an increment equal to
ymin and y=f(basic cycle length-141)
defined the restitution function f shifted to the right by an increment
equal to ymax. In this example, a difference
between ymax and ymin of
7 ms was associated with a 59-ms range of possible APD values during
irregular dynamics. Reducing the difference between
ymax and ymin to 2 ms
decreased the range of APD to 10 ms (not shown).
With respect to the second objective, irregular dynamics occurred, both
experimentally and in the model, between regions of 1:1 and 2:2
locking. Stable 1:1 dynamics were expected22 and found
when the line indicating basic cycle length intersected the restitution
curve at only one point, the fixed point, and the fixed point was on a
segment whose absolute slope was <1. Examples of convergence onto the
fixed point for three different pacing cycle lengths are shown in Fig 5A. Increasing ymax increased the
slope of the terminal segment of the ascending region of APD
restitution (the region of intersection with a basic cycle length of
138 ms in Fig 5A). As a result, the pacing cycle length at which 1:1
locking was replaced by irregular dynamics increased (compare panels A
and C of Fig 4).
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A geometric method was found for determining the basic cycle length
range supporting 2:2 behavior. The essence of this method (see Fig 5B)
was to first find the two points of intersection between a line
described by y=x+
and the restitution function
and then determine the product of the slopes of the function at those
two points. If the absolute value of the product was <1, iterations of
the function at the corresponding cycle length would eventually lead to
2:2 behavior.22 In the example in Fig 5B, the equation for
the line was y=x+115, and the intersections of
this line with the restitution function lay on segments whose slopes
were 9 and -0.1. The limits of the cycle length over which 2:2 locking
occurred could then be found by moving the
y=x+ line up or down until the products of the
slopes exceeded 1 and then finding the corresponding cycle length. For
example, the largest value associated with 2:2 locking was 120. The
line y=x+120 intersected the restitution function
at coordinates 15,135 and -8.125, 111.875. The sum of the x
coordinate of one intersection point and the y coordinate of
the other intersection point (ie, 15+111.875 or -8.125+135) gave the
corresponding cycle length of 126.875. Decreasing
ymin increased the slope of the descending
region of APD restitution (the upper region of intersection with the
line described by y=x+115 in Fig 5B).
Consequently, the pacing cycle length at which irregular dynamics was
replaced by 2:2 locking decreased (compare panels A and C of Fig 4).
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Discussion |
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The induction of complex behavior in ventricular muscle required the presence of a steeply sloped ascending region of APD restitution and a critical point. Similar requirements for complex behavior were found previously in Purkinje fibers and in ventricular muscle exposed to heptanol.17 The critical point in the latter study was a local minimum resulting from the induction of significant latency to activation during phase 3 of the action potential. In contrast, the critical point in the present study was a local maximum corresponding to an overshoot of APD following terminal repolarization, with a minimal contribution of latency. The presence of a steeply sloped function and a critical point in the present study and in the study by Chialvo et al17 was associated, as expected,22 with certain qualitative similarities in dynamic behavior (eg, period-doubling bifurcations preceding chaotic dynamics). However, these dynamically similar phenomena most likely resulted from quite different ionic mechanisms, in that the latency associated with activation at short diastolic intervals probably involved delayed activation of partially recovered Ca2+ current in the presence of decaying Ik and Ik1,23 whereas the overshoot of APD at longer diastolic intervals is believed to reflect potentiation of the Ca2+ current,13 24 25 26 secondary to voltage-dependent phosphorylation of the
1 subunit.27
Biphasic restitution also was associated with period-doubling
reversals,28 a dynamic behavior not previously reported
for periodically forced cardiac tissue. The development of
period-doubling reversals required a region of shallow slope in the APD
restitution function at diastolic intervals longer than those at which
the overshoot of APD occurred (ie, to the right of the overshoot in
Figs 3B and 5). As the pacing cycle length was shortened, aperiodic
activity was replaced by periodic activity when iterations of the map
stabilized on regions whose slope values, when multiplied, had an
absolute value of <1. Such a period-doubling reversal could be
eliminated by increasing the slope of APD restitution at longer
diastolic intervals. In this case, aperiodic activity was followed only
by 2:1 block.
Period-doubling bifurcations and aperiodic behavior also have been elicited in Purkinje fibers and ventricular muscle having steeply sloped, but monotonic, restitution functions, by exploiting the presence of supernormal excitability.23 29 30 The discontinuities in excitability associated with supernormality introduce nonmonotonicity into an otherwise monotonic relation between diastolic interval and APD. Although supernormal excitability can be demonstrated experimentally over a narrow range of diastolic intervals in normal ventricular muscle,31 32 the potential contribution of supernormality to the development of complex dynamics was excluded in the present study, as in the study of Chialvo et al,17 by driving the preparations by use of stimuli whose intensities were at least twice the late diastolic threshold.
For reasons that have been discussed in detail previously,1 2 3 4 5 20 33 34 complex rate-related changes in APD, such as those arising from biphasic restitution, are likely to alter refractoriness and thereby contribute to the initiation, perpetuation, or termination of reentrant arrhythmias. In addition, changes in APD might affect wave propagation secondary to concomitant alterations of APA and dV/dtmax. For example, the relatively small range of APD during 8:8 locking was associated with a large variation in APA. This observation may pertain to the positive association between T-wave alternans, as recorded on the surface ECG, and the subsequent development of sudden death,4 35 in that apparent T-wave alternans resulting from 2:2 or 4:4 locking of APD might be associated with larger than expected alterations of APA and conduction velocity. Moreover, the association of T-wave alternans with more complex behavior may not be straightforward, in that a typical period-doubling route to chaos would not necessarily be expected for ventricular muscle in which the restitution of APD was biphasic; ie, APD alternans may either precede or follow more complex dynamics.
From the computer model, we found that by using geometric arguments key features of the bifurcation diagram could be approximated without iterating the restitution function. For example, the range of possible APD during irregular dynamics was bounded by the following two equations: y=ymax and y=f(BCL-ymin), where BCL is basic cycle length. Accordingly, the greater the difference between ymax and ymin, ie, the greater the amplitude of the overshoot, the larger the range of APD during irregular dynamics. Similarly, the larger the amplitude of the overshoot, the greater the range of cycle lengths over which the slopes of the ascending and descending regions of the restitution function were sufficiently steep to support irregular dynamics. From these results, it seems possible that the presence or absence of biphasic restitution of APD in different regions of myocardium, as observed in the human heart,13 16 could predispose to the development of marked rate-dependent heterogeneity of refractoriness.
Although the results generated by the computer model agreed reasonably well in most respects with those observed experimentally, the piecewise linear approximation may have failed to reproduce certain behavior. For example, successive period-doubling bifurcations leading to irregular behavior did not occur in the piecewise linear model, whereas they were observed over small (<1-ms) ranges of pacing cycle length in a continuous model (not shown). Both of these results are expected, in that the piecewise linear modeling produces a discrete and finite set of slope values, whereas the continuous model allows for a dense set of slopes. Experimentally, the apparently sharp transition from 1:1 locking to irregular dynamics and from irregular dynamics to 2:2 locking may have been preceded by period-doubling bifurcations or reversals that were undetected when the 1-ms decrements in pacing cycle length were used.
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Acknowledgments |
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This study was supported by National Institutes of Health grants HL-40800 and HL-39707 (Dr Gilmour) and by a National Science Foundation Presidential Young Investigator Award (Dr Otani). Dr Watanabe is a Howard Hughes Medical Institute Predoctoral Fellow. We thank D.R. Chialvo and J. Jalife for helpful discussions, V. Meyers-Wallen and P. Nathanielsz for providing the fetal tissue, and E.A. Peet for technical assistance.
Received August 11, 1994; accepted January 4, 1995.
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